Blender - Some of the bones in armature moving mesh
The issue is that you have skinned your mesh before symmetrizing your rig.
Result: the newly created part of the rig isn't skinned properly to the mesh. And I also guess your mesh may not have vertex groups for those new bones.

First, make sure you have vertex groups with the same names as each deforming bone.

Then, you'll have to edit the weights of your mesh in order to assign the right side properly to the right side bones. Either use weight paint mode, either edit mode via the vertex groups pannel.